Which of the following could be a factor of both 2x+1 and 4x-50 for so

Two ways.

I. Remainders

Now 2x+1 is odd, so it will never have a factor 2. Thus we can remove 2 from 4x-50 or 2(2x-25).
So (2x+1) and 2x-25 will still have that common factor we are looking for.

Whatever divides both 2x+1 and 2x-25 will surely divide (2x+1+2x-25) or 4x-24 and (2x+1)-(2x-25) or 26.
So the common factor has to be 2, 13 or 26.
But we know it cannot be even, so 13 is the answer.


II. Options
Discard 2 as 2x+1 is always odd for any integer value of x.
Common factor =3: if x is 4, then 2x+1=9, but 4x-50=4*4-50=-34, which is not divisible by 3
Common factor =7: if x is 3, then 2x+1=7, but 4x-50=4*3-50=-38, which is not divisible by 7
Common factor =13: if x is 6, then 2x+1=13, and 4x-50=4*6-50=-26, which is divisible by 13
Common factor =19: if x is 9, then 2x+1=19, but 4x-50=4*9-50=-14, which is not divisible by 19

13 is the only option which divides both the expressions


D
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I don't quite follow the last bit of this solution. If you plug in 13 (as the value of x) to each, you get 27 and 2, right? And those have nothing in common. (Note that if you had actually gotten 3 as a common factor, the answer would not be D, since the question asks for the common factor, not the value of x.)

We could continue up to 19, when plugging in gives 39 and 26, and there is a common factor of 13. And that's actually proof that the answer is D.

However, even though we did eventually find the answer this way, this approach is very tedious and mistake-prone, and it may not work for any answer choice, since the answers are not values of x. So it's much better to find a method other than number-picking.
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Solution

• We can rewrite \(4x-50\) as \(4x -2+2-50 = 4x +2 – 52 = 2*(2x+1) – 52\).
• Now, we can observe, if a number is a factor of \( 2x+1\), it must be a factor of \(2*(2x+1)\).

o So, we need to focus only on \(2x+1\) and 52 and find that from the given answer options which one could be common factor of both 2x+1 and 52.
• Let’s prime factorize 52, \(52 = 2*2*13\)

Now, if we look at the answer options, we can straightaway eliminate Options B, C and E, since they are not a factor of 52.
Let’s check remaining two Options (i.e. A and D) and eliminate the wrong option:

• Option A: 2

o For all integer \(x\), \(2x+1\) is odd. So, 2 cannot be a factor of \(2x+1\).
o Hence, we can eliminate answer Option A too.

Thus, the correct answer is Option D.
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We can start backsolving to crack it easily...
When we substitute 2 in 2x+1 and 4x-50 we get 5 and -42 which have nothing in common
Next 3 gives 7 and -38 which have nothing in common
With 7 we get15 & -22 which have nothing in common. With 13 we get 27 & -24 which have 3 as the common factor. Hence D

Posted from my mobile device

This must have been already discussed.

The easiest is to discard 2, as 2x+1 will always be odd when x is an integer.
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No problem! Your revised solution works, and is very similar to GMATWhizard's.

For my own solution, which takes a different approach, check out my video on this subject, in conjunction with GMAT Club (this question is presented at the 1:09.40 mark, and the solution starts about two minutes after that):

Thinking Like a Number Theorist
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Values of both 2x+1 and 4x-50 can be calculated for different values of x

x; 2x+1; 4x-50
1; 3; -46; No common factor
2; 5; -42; No common factor
3; 7; -38; No common factor
4; 9; -34; No common factor
5; 11; -30; No common factor
6; 13; -26; Common factor is 13

IMO D

Thanks Anthony for pointing out the error...I have inadvertently calculated 2x-50 instead of 4x-50..
Please see this solution.
The question asks which of the 5 solutions is a factor of both 2x+1 and 4x-50...
Let us write 4x-50 as 2[(2x+1)-26]
Let us say 2x+1=y
So we need to see whether any of the given numbers divide both y and 2(y-26).
2x+1 is obviously odd. So 2 won't divide y. Hence A is not an option.
Now both 3 and 7, even if they divide y, they won't divide 26. So they won't divide 2[(2x+1)-26] that is 4x+50. So B and C are not the answers.
Coming to D, if 13 were to divide 2x+1, since 13 divides 26, 13 will divide y=2[(2x+1)-26] as it divides both 2x+1 and 26. So D is the answer.
If 19 were to divide 2x+1, it won't divide 4x+50 as 19 doesn't divide 26...
Hence D is the only option. Hope this clarifies.
Sorry once again, for the confusion in my earlier post. And thanks for bringing it to my attention.

Posted from my mobile device

lets go by option elimination

of the given numbers one is odd and other is even so factor must definitely be odd
so option A is eliminated

now consider option B

2x+1 =3k (let us consider 2x+1 has 3 as a factor)
we get x=3k-1/2 now we substitute the value of x in 4x-50 an see that whether it is a fctor of 3 or not
4(3k-1)/2 -50=6k-52 which is not a multiple of 3

similarly consider option C
we get 14k-52 finally which is not a multiple of 7

now option D
we get 26k-52 which is indeed a multiple of 13

and option E gives 38k-52 which is not a multiple of 19

Hence choice D

which of the following could be a factor of both 2x+1 and 4x-50 for some integer x?
a)2
b)3
c)7
d)13
e)19
what I did - I tried to find the value of 2x+1 and 4x-50 by solving these two equations and I got them as 52 and -52
In the options, I was confused between 2&13
however, the correct option is 13. How do we reject 2 and select 13

well if you notice
2x+1 will give odd value for any x and
4x-50 will give even value for any value of x ; this way you can negate 2 or any even option..
now for 2x+1 & 4x-50 substitute values x with ( 1,2.. 6) & check whether the result is a factor of given options ; we see that its a factor of 13 at x= 6 also it works with 4x-50 ;
13 is correct
angelthomas

This is how I tried to solve this problem.
Please let me know if you find problems with the solution.
If 2x+1 and 4x-50 have a common divisor then 2*(2x+1) and 4x-50 shall also have the common divisor.
Now, if you add or subtract multiples of a number, the result should also be a multiple of that number.
so, subtracting 2*(2x+1) and 4x-50 we get 52, which is divisible by 2 and 13.
But as we know 2x+1 is odd, it cannot be 2. Therefore, 13 is the answer.

Best way to solve is by hit and trial

Select some logical value of X as 15, 16,17,18,19

when X is 19 then 2X+1 = 39 and 4X-50 is 26. Common factor is 13 which is the answer.

Ans D.

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